Volume 12, Issue 1

We focus on the stability analysis of two types of discrete dynamic models: a discrete dynamic equation and a discrete dynamics system consisting of two equations with mutualistic interaction given by x n + 1 = a + b x n λ − ( x n − 1 + x n − k ) c + x n − 1 + x n − k {x}_{n+1}=a+\frac{b{x}_{n}{\lambda }^{-\left({x}_{n-1}+{x}_{n-k})}}{c+{x}_{n-1}+{x}_{n-k}} and x n + 1 = a 1 + b 1 y n λ − ( y n − 1 + x n − k ) c 1 + y n − 1 + x n − k , y n + 1 = a 2 + b 2 x n λ − ( x n − 1 + y n − k ) c 2 + x n − 1 + y n − k , {x}_{n+1}={a}_{1}+\frac{{b}_{1}{y}_{n}{\lambda }^{-({y}_{n-1}+{x}_{n-k})}}{{c}_{1}+{y}_{n-1}+{x}_{n-k}},\hspace{1.0em}{y}_{n+1}={a}_{2}+\frac{{b}_{2}{x}_{n}{\lambda }^{-\left({x}_{n-1}+{y}_{n-k})}}{{c}_{2}+{x}_{n-1}+{y}_{n-k}}, respectively, where k ∈ { 2 , 3 , … } k\in \left\{2,3,\ldots \right\} , the constants a , a 1 , ≥ 0 a,{a}_{1},\ge 0 , b , b 1 > 0 b,{b}_{1}\gt 0 , and c , c 1 ≥ 0 c,{c}_{1}\ge 0 be the initial densities, finite rate of increase and the limiting constant associated with the density of the species, respectively, and a 2 ≥ 0 {a}_{2}\ge 0 , b 2 > 0 {b}_{2}\gt 0 , and c 2 ≥ 0 {c}_{2}\ge 0 be the initial densities, finite rate of increase, and the limiting constant associated with the density of the mutually interacting species, respectively. k k , a positive integer represents a time delay in the system and λ ≥ 1 \lambda \ge 1 shows a decay factor based on the sum of two past time step population densities. Our main objective is to understand the impact of mutualistic interactions on the stability of discrete dynamic systems. To illustrate the boundedness and stability of these models, we also provide animated plots and bifurcation diagrams.

The outbreak of coronavirus disease 2019 (COVID-19) has been declared a pandemic by the world health organization on March 11, 2020,. Here, a nonlinear mathematical model is proposed and analyzed to study the spread of coronavirus disease in a human habitat. In modeling the dynamics, the total population is divided into five subclasses: susceptible population, asymptomatic infective population, symptomatic infective population, recovered population, and vaccinated population. It is assumed that the disease is transmitted directly from infectives. It is further assumed that due to the effect of media, susceptible individuals become aware about the disease and avoid contact with the infectives. The analysis of the model is performed using the stability theory of differential equations. Furthermore, conditions that influence the persistence of the system are obtained. We have also conducted numerical simulations to validate the analytical results. The model analysis shows that with an increase in media awareness, the spread of coronavirus disease decreases with a decrease in the number of infective populations.

In this article, we present a susceptible, exposed, infected, hospitalized and recovered compartmental model for COVID-19 with vaccination strategies and mask efficiency. Initially, we established the positivity and boundedness of the solutions to ensure realistic predictions. To assess the epidemiological relevance of the system, an examination is conducted to ascertain the local stability of the endemic equilibrium and the global stability across two equilibrium points are carried out. The global stability of the system is demonstrated using Lyapunov’s direct method. The disease-free equilibrium is globally asymptotically stable when the basic reproduction number (BRN) is less than one, whereas the endemic equilibrium is globally asymptotically stable when BRN is greater than one. A sensitivity analysis is performed to identify the influential factors in the BRN. The impact of various time-dependent strategies for managing and regulating the dynamic transmission of COVID-19 is investigated. In this study, Pontryagin’s maximum principle for optimal control analysis is used to identify the most effective strategy for controlling the disease, including single, coupled, and threefold interventions. Single-control interventions reveal physical distancing as the most effective strategy, coupled measures reduce exposed populations, and implementing all controls reduces susceptibility and infections.

In this research, we have introduced compartments for asymptomatic and symptomatic individuals, along with reduced susceptibility, as key factors defining our investigation. The study is carried out in diverse scenarios, considering them as crucial for the essential generation number of the model, set at 3.18( r 0 > 1 {r}_{0}\gt 1 ). The persistent reproduction differential method was used to explore the impact of continuous adrenocorticotropic hormone (ACTH) administration on the global gene expression in primary cultures of both fetal and adult adrenal cells. The study also investigates ACTH’s genetic effects on both adult and fetal human adrenal cells. The conclusion of this study is demonstrated through relevant and correct medical applications.

This work builds machine learning models for the dataset generated using a numerical model developed on an idealized human artery. The model has been constructed accounting for varying blood characteristics as it flows through arteries with variable vascular properties, and it is applied to simulate blood flow in the femoral and its continued artery. For this purpose, we designed a pipeline model consisting of three components to include the major segments of the femoral artery: CFA, the common femoral artery and SFA, the superficial artery, and its continued one, the popliteal artery (PA). A notable point of this study is that the features and target variables of the former component pipe form the set of features of the latter, thus resulting in multicollinearity among the features in the third component pipe. Thus, we worked on understanding the effect of these correlated features on the target variables using regularized linear regression models, ensemble, and boosting algorithms. This study highlighted the blood velocity in CFA as the primary influential factor for wall shear stress in both CFA and SFA. Additionally, it established the blood rheology in PA as a significant factor for the same in it. Nevertheless, because the study relies on idealized conditions, these discoveries necessitate thorough clinical validation.

In this manuscript, we examine the blood alcohol model to investigate the dynamics of alcohol concentration in the human body. The classical model of blood alcohol concentration is converted into the fractional model by using Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo derivatives. The existence and uniqueness theory for the model’s solution is constructed using the Banach fixed point theory. Also, the stability of the solution is established by Ulam-Hyers conditions. For the numerical simulation of the considered model, the Adams-Bashforth method with a two-step Lagrange polynomial is used and the numerical solution of the model with three different derivatives is presented in the tabular and graphical form. The comparison between the exact solution and observed solution is made by root mean square technique which is found to be in good agreement. Finally, the results from the three fractional derivatives are also compared with the exact data, which revealed that the CF fractional derivative performs better than the other two fractional derivatives.

In this study, a nested multi-scale mathematical model is proposed and studied for COVID-19 viral infection. The well-posedness of the multi-scale model is discussed, followed by the stability analysis of the equilibrium points. The proposed model admits two equilibrium solutions: the infection-free equilibrium and the infected equilibrium. The infection-free equilibrium point is shown to be globally asymptotically stable provided that the value of the basic reproduction number is less than unity. When the value of the basic reproduction number exceeds unity, a unique infected equilibrium exists, and the system is found to undergo a forward (trans-critical) bifurcation. Two-parameter heat plots are performed to identify the parameter combinations for the stability of the admitted equilibrium points. From the sensitivity analysis, the transmission rate, infection rate, and birth rate were found to be most sensitive to the basic reproduction number. The influence of the within-host sub-model parameters on the between-host sub-model variables is numerically illustrated. The spread of infection in a community is shown to be influenced by the within-host sub-model parameters, such as the production of viral particles, the clearance rate of infected cells by the immune system, and the clearance rate of viral particles by the immune system. The comparative effectiveness of three health interventions (antiviral drugs, immunomodulators, and generalized social distancing) for COVID-19 infection is examined using the reproductive number as an indicator of the effectiveness of these interventions. The results from the comparative effectiveness study suggest that a combined strategy of antiviral drugs, immunomodulators, and generalized social distancing would be the best strategy to implement to contain the spread of infection in the community.

Although heart disease stands as a prominent contributor to worldwide deaths, not all individuals affected by it ultimately fall prey to its effects. Timely diagnosis and effective treatment can offer those with heart conditions a high-quality life in their later years. Consequently, early disease detection using accessible medical data has been a central goal for researchers in recent decades. Traditionally, researchers relied on statistical tools for this purpose. However, machine learning algorithms, especially classification models, have gained prominence with the growing accumulated data. These algorithms have shown promise in predicting heart disease based on individual data. Our study employed various classification algorithms to predict heart disease incidence using the available dataset. We prioritized model reliability by incorporating the conformal classifier. Our results have shown that boosting algorithms, such as XGBoost and CatBoost, demonstrated exceptional performance with promising metrics. These models identified chest pain type and ST segment slope as crucial indicators of heart disease. Boosting algorithms exhibited a compelling combination of broad coverage and a small prediction set size, making them well-suited for heart disease prediction. Furthermore, we employed explainable artificial intelligence-boosting algorithms to enhance the interpretability of our predictions.

Harmful algal blooms are one of the major threats to aquatic ecosystem. Some phytoplankton species produce toxins during algal bloom and affect other aquatic species as well as human beings. Thus, for the conservation of aquatic habitat, it is much needed to control such phenomenon. In the present study, we propose a mathematical model of toxin-producing phytoplankton and zooplankton species, which follows the Holling Type III functional response. We consider the effect of prey refuge and harvesting on both the species. Boundedness of the proposed model, existence of equilibria, and their stability have been discussed analytically. We also discuss the optimal harvesting policy and existence of bionomic equilibrium. The numerical simulation has also been performed. We identify the control parameters that are responsible for the system dynamics of the model. The parameter prey refuge has a great impact on the dynamics of the model system. Higher value of prey refuge leads to the stable dynamics. Also, the growth rate of phytoplankton acts as a control parameter for the dynamics of the model. The higher value of growth rate of phytoplankton is responsible for oscillatory behavior.

During the recent Coronavirus disease (COVID-19) pandemic, different parts of the globe faced indefinite lockdowns. To maintain the lockdown measures, government authorities deployed security forces and police. The primary role of the police is to monitor the movement of the people and advise/guide them to follow the lockdown. In the current work, we propose a nonlinear mathematical model to study the role of police and security personnel in controlling COVID-19. It is observed that the proposed model possesses two equilibria, namely, trivial and non-trivial. We performed the stability analysis of the non-trivial equilibrium point by using the compound matrix technique. Finally, we perform a detailed numerical simulation to investigate the feasibility of the theoretical results. The current study demonstrates that police forces played a significant role in controlling the pandemic.

In this article, we present a novel approach for robust state estimation in linear uncertain descriptor systems, with a specific focus on the cardiovascular system (CVS) as a case study. Descriptor systems, which extend traditional state-space representations by incorporating algebraic constraints, present significant challenges in observer design, especially under conditions of uncertainty. To address these challenges, we propose the design of a functional observer using the H ∞ {H}_{\infty } filter approach, capable of managing uncertainties arising from both external disturbances and variations in system parameters. The observer is designed using a unique linear matrix inequality framework, which ensures the stability and performance of the estimation error system. The convergence of the estimation error is rigorously analyzed using the Lyapunov method. The effectiveness of the proposed method is demonstrated through simulations on a CVS model, highlighting its potential to reliably estimate unknown states in the presence of uncertainties.

In this study, we formulate an eight-compartment mathematical model with vaccination as one of the compartments to analyze the dynamics of COVID-19 transmission. We examine the model’s qualitative properties, such as positivity and boundedness of solutions, and stability analysis of the illness-free equilibrium with respect to the basic reproduction number. We estimate ten significant parameters and also compute the magnitude of the basic reproduction number for India by fitting the proposed model to daily confirmed and cumulative confirmed COVID-19 cases in India. Sensitivity analysis with respect to basic reproduction number is conducted, and the main parameters that impact the widespread of disease are determined. We further extend this model to an optimal control problem by including four non-pharmaceutical and pharmaceutical intervention measures as control functions. Our numerical results show that the four control strategy has greater impact than the three control strategies, two control strategies, and single control strategies on reducing the dynamics of COVID-19 transmission.

This article addresses the pressing issue of pest outbreaks in India, which poses significant challenges for farmers and ecologists. A novel system is proposed for effective control that leverages natural enemies. Here, the pests are classified into juveniles and mature individuals, further categorized as susceptible or infected. The study introduces harvesting, incorporating external efforts and natural phenomena, in a pest-epidemic prey–predator system featuring a prey-stage structure. The model reveals three equilibria: trivial, boundary (indicating the absence of natural enemies), and interior equilibria. Notably, the trivial equilibrium is consistently unstable. As demonstrated by stability analysis, the survival or extinction of natural enemies hinges on control variables, including the harvesting rate, disease transmission rate, and natural death rate. Local stability is assessed using the Routh–Hurwitz criterion, while global stability is explored through the Lyapunov method. Furthermore, optimal control theory and Pontryagin’s maximum principle are applied for model optimization, unveiling crucial optimality conditions and determining the optimal harvesting rate for susceptible mature prey. Numerical computations validate theoretical insights, offering valuable guidance for formulating policies that optimize the control of susceptible adult pests within a disease-induced pest-natural enemy system, ensuring sustained cost-effectiveness.

We observed the existence of periodic orbits in immune network under transitive solvable Lie algebra. In this article, we focus to develop condition of maximal Lie algebra for immune network model and use that condition to construct a vector field of symmetry to study nonlinear pathogen model. We used two methods to obtain analytical structure of solution, namely normal generator and differential invariant function. Numerical simulation of analytical structure exhibits correlated periodic pattern growth under spatiotemporal symmetry, which is similar to the linear dynamical simulation result. We used Lie algebraic method to understand correlation between growth pattern and symmetry of dynamical system. We employ idea of using one parameter point group of transformation of variables under which linear manifold is retained. In procedure, we present the method of deriving Lie point symmetries, the calculation of the first integral and the invariant solution for the ordinary differential equation (ODE). We show the connection between symmetries and differential invariant solutions of the ODE. The analytical structure of the solution exhibits periodic behavior around attractor in local domain, same behavior obtained through dynamical analysis.

In this study, we present a variational multiscale stabilized finite element method for steady-state incompressible fluid flow under magnetic forces. In particular, an algebraic approach of approximating the subscales has been considered, and then, the stabilization parameters are derived using Fourier analysis. The proposed scheme is used to trace the blood flow dynamics in complex arterial vessels under multiple pathological conditions. We examine the pressure and stress distribution in addition to the flow pattern to assess the criticality of the diseased condition.

As per the World Health Organization’s (WHO’s) suggestions, personal protection via adopting precautionary measures is one of the most effective control aspects to avoid Zika infection in the absence of suitable medical treatment. This personal protection further can be enhanced and explored by propagating information about disease prevalence. Therefore, in this study, we wish to see the effect of information on Zika transmission by formulating a compartmental mathematical model that quantifies the effect of an individual’s behavioral response as self-protection due to information. Furthermore, the basic reproduction number was calculated using the next-generation matrix technique. The model analysis was carried out to determine the local and global stability properties of equilibrium points. In addition, the model shows the occurrence of forward bifurcation when the reproduction number crosses unity. To understand the impact of various model parameters, we conducted a sensitivity analysis using both the normalized sensitivity index and the partial rank correlation coefficient methods. Moreover, we performed numerical simulations to assess the influence of important parameters on the model’s behavior for Zika prevalence. Our study accentuates that as information-induced self-protection increases, the prevalence of Zika infection will be at a very minimum level, and this observation is in line with WHO suggestions.

In this work, we propose a hybrid impulsive prey-predator-mutualist model on nonuniform time domains. We have investigated the permanence/persistence results for the proposed model using the comparison theorems of impulsive differential equations and some dynamic inequality on the nonuniform time domains. In addition, we have established certain necessary requirements for the uniform asymptotic stability of the almost periodic solution and global attractivity of the proposed model. Furthermore, we provide several numerical examples on nonuniform time domains with computer simulation to demonstrate the viability of the results of the acquired analytical work.

Urbanization has been a growing trend for decades, but the Coronavirus disease 2019 (COVID-19) pandemic has accelerated this trend in several ways. In many countries, the pandemic has prompted significant investments in urban health infrastructure, including hospitals, clinics, and public health facilities. Also, there are many studies suggesting that COVID-19, urbanization, and chronic diseases are interconnected in several ways, and addressing one aspect requires a comprehensive approach that considers the interconnectedness of these factors. To date, there are no studies depicting COVID-19 as a cause of emerging chronic diseases within the urban population. To address this sensitive issue, we propose a mathematical model in which we divide the total human population into three compartments: susceptible, diseased, and recovered. We have obtained the equilibrium points of the model and the conditions for their local stabilities. A suitable Lyapunov function has been used to derive the condition for the global stability of the nontrivial equilibrium point. A detailed numerical simulation of the model has been carried out, which is followed by sensitivity analysis of the model system.

The emergence and dynamic prevalence of genetic disorders and infectious diseases with mutations pose significant challenges for public health interventions. This study investigated the parameter estimation approach and the application of the dynamic state-space Markov modeling of these conditions. Using extensive simulations, the model demonstrated robust parameter estimation performance, with biases and mean-squared errors decreasing as sample size increased. Applying the model to COVID-19 data revealed distinct temporal patterns for each variant, highlighting their unique emergence, peak dominance, and decline or persistence trajectories. Despite the absence of clear trends in the data, the model exhibited a remarkable accuracy in predicting future prevalence trends for most variants, showcasing its potential for real-time monitoring and analysis. While some discrepancies were observed for specific variants, these findings suggest the model’s promise as a valuable tool for informing public health strategies. Further validation with larger datasets and exploration of incorporating additional factors hold the potential for enhancing the model’s generalizability and applicability to other evolving diseases.

Approximately 37.9 million people living with HIV (PLWH) are at risk of severe consequences from COVID-19. Urgent development of tailored treatments for PLWH, who have historically been excluded from vaccine trials, is crucial. The present study introduces some modified fulleropyrrolidine derivatives with chalcogen atoms (O, S, or Se) and hydroxymethylcarbonyl (HMC) groups to target 11 single and double HIV-1 protease (HIV-PR) mutations and the main protease of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2 M pro ). The inhibitory activities of these derivatives are computationally examined through molecular docking, molecular dynamic simulations for 200 ns, and Lipinski’s rule of five (RO5). Fourier-transform infrared spectroscopy spectra and thermodynamic properties are calculated and analyzed using Density Functional Theory B88-PW91 method. The results indicate that the suggested O-compounds obey three parameters of the RO5 and HMC forms hydrogen bonds with studied viral proteases. Compounds with O and S additives display a high binding affinity with negative binding energy values for HIV-PR mutations (A71V-I84V, V77I-I84V, and I84V-L90M) and SARS-CoV-2 M pro . The compounds with S and Se additives shift to lower frequencies of the major vibrational bands. Specifically, compound 1, with two oxygen additives, emerges as the most effective in inhibiting both HIV-PR mutations and SARS-CoV-2 M pro .

The open question whether quantum field theories can be used in biological systems will be addressed in this study. Quantum effects were reported for certain biological systems like the magnetic orientation sense in migratory birds. In quantum field theories it is possible to derive effective quantum field theories with welded tangleoids including braid relations that describe composite particles based on the dynamics of microscopic particle where these composite particles are made of. We observe that the generators of the tangleoid category that will depict the Feynman graphs are X X for a scattering vertex. This arises after carrying out the integral over bosonic fields A μ {A}_{\mu } in the partition function (D) that will lead to an four-valent interaction vertex generated by a quartic term in the fermionic fields. With X + {X}_{+} and X − {X}_{-} we will depict order changes like that regarding time orderings. Ordinary propagators are depicted by ∪ \cup and ∩ \cap . Finally, the generators ! \! and ¡ \hspace{0.1em}\text{¡}\hspace{0.1em} come into play if other foreign fields are picked up.

Infectious illnesses like hepatitis place a heavy cost on global health, and precise mathematical models must be created in order to understand and manage them. The Adomian decomposition method (ADM) and an optimal control strategy are utilized to solve a fractional-order hepatitis model in this research. By adding fractional derivatives to account for memory effects and non-integer order dynamics, the fractional-order model expands the conventional compartmental model to take into account the complexity of hepatitis dynamics. The fractional-order hepatitis model is resolved using the ADM, a powerful and effective analytical approach. This approach offers a series solution that converges quickly, enabling the model’s precise analytical solution to be derived. To identify crucial criteria and enhance control mechanisms for the management of hepatitis, an optimal solution strategy is also introduced. The optimization procedure tries to lessen the disease’s spread and its negative effects on public health. We can find the best interventions, immunization schedules, and treatment regimens to effectively reduce the hepatitis pandemic by integrating the ADM solution with an optimization framework. The findings of this study show that the suggested method may be used to solve the fractional-order hepatitis model and optimize control measures. The analytical solution produced by ADM offers important insights into the underlying dynamics of hepatitis transmission, and the optimization process produces suggestions that public health professionals and politicians may put into practice. In the end, this research presents a promising direction for improving disease control efforts in a fractional-order context and contributes to a deeper understanding of hepatitis epidemiology. The importance of this method is that it gives solutions that coincide with that obtained using the numerical approach.

In this article, we study the role of awareness and its impact on the control of infectious diseases. We analyze a susceptible-infected-recovered model with a media awareness compartment. We find the effective reproduction number R 0 {R}_{0} . We observe that our model exhibits transcritical forward bifurcation at R 0 = 1 {R}_{0}=1 . We also performed the sensitivity analysis to determine the sensitivity of parameters of the effective reproduction number R 0 {R}_{0} . In addition, we study the corresponding optimal control problem by considering control in media awareness and treatment. Our studies conclude that we can reduce the rate of spread of infection in the population by increasing the treatment rate along with media awareness.

Alzheimer’s disease is one of the most prevalent types of dementia worldwide. It is caused by the accumulation of amyloid-beta (Aβ) plaques in the brain, disrupting communication pathways and memory. Microglia and astrocytes act as the immune system of the brain, clearing Aβ plaque deposits. However, these cells can lose effectiveness when Aβ plaque accumulation exceeds normal limits, leading to inflammation induced by proinflammatory cytokines. One type of treatment involves anti-Aβ drug therapy. Anti-Aβ drugs are believed to have the ability to reduce Aβ plaque deposits effectively. The mechanism of Aβ plaque accumulation can be explained by ordinary differential equations describing the growth of Aβ monomers. In this study, we aimed to develop a new mathematical model to elucidate the role of the immune system and drug therapy in reducing Aβ plaque deposits. Based on the simulation results, we conclude that the use of anti-Aβ drug therapy can decrease the concentration of Aβ plaque deposits, and the effective treatment duration for Alzheimer’s patients is estimated to be approximately 4 months starting from the time the drug was first administered.

The objective of this article is to study the compartmental modeling approach for the prediction of unreported cases of coronavirus disease 2019 by considering six compartments. Our model is described by a system of six ordinary differential equations with initial conditions. The basic properties of solution of the model are established. The model is shown to have two equilibrium points, i.e., the disease-free and endemic equilibrium points. The basic reproduction number R 0 {R}_{0} is derived by the next-generation matrix method. Stability analysis is carried out in the study. Furthermore, sensitivity analysis is also performed to identify the impact of important parameters that significantly affect R 0 {R}_{0} . Numerical simulations provide a good approximation model for COVID-19, which will be utilized to investigate future pandemic with similar nature of spread as COVID-19 and estimate the number of unreported cases worldwide.

Close-contact places such as long-term facilities have been found to be high-risk and high-morbidity places in the United States during COVID-19 outbreaks. This could be due to the presence of vulnerable resident population, frequent contacts of residents with visitors, staff working at multiple facilities, and potential contaminated surfaces at these facilities. Here, we model close-contacts places to evaluate the role of different transmission pathways of COVID-19 in the presence of adaptable interventions. The model captures a coupled dynamics between three subpopulations (staff, residents, and visitors) and incorporates infection from infectious individuals and through the environment. By using parameterization of the models via real data from facilities in the United States, we identify and quantify the impact of duration and choice of interventions in real time for subpopulations to mitigate disease burden. We study the trade-off between disease burden and prevention cost using cost-effectiveness analysis. It was found that the specific time interval in which an intervention is included has an important effect when maximizing the effectiveness while minimizing the costs. We find that the presence of super-spreader healthcare workers contribute to a significantly higher peak of the number of infected cases compared to similar situation in the presence of super-spreader residents. Considering two nonpharmaceutical interventions: the use of face masks and cleaning surfaces, we observe that the later is an optimum intervention at the beginning of the system’s evolution, while wearing masks becomes an optimum strategy once the infection is widely established and population behavior changes.

Variational implicit solvation models (VISMs) have gained extensive popularity in the molecular-level solvation analysis of biological systems due to their cost-effectiveness and satisfactory accuracy. Central in the construction of VISM is an interface separating the solute and the solvent. However, traditional sharp-interface VISMs fall short in adequately representing the inherent randomness of the solute–solvent interface, a consequence of thermodynamic fluctuations within the solute–solvent system. Given that experimentally observable quantities are ensemble averaged, the computation of the ensemble average solvation energy (EASE)–the averaged solvation energy across all thermodynamic microscopic states–emerges as a key metric for reflecting thermodynamic fluctuations during solvation processes. This study introduces a novel approach to calculating the EASE. We devise two diffuse-interface VISMs: one within the classic Poisson–Boltzmann (PB) framework and another within the framework of size-modified PB theory, accounting for the finite-size effects. The construction of these models relies on a new diffuse interface definition u ( x ) u\left(x) , which represents the probability of a point x x found in the solute phase among all microstates. Drawing upon principles of statistical mechanics and geometric measure theory, we rigorously demonstrate that the proposed models effectively capture EASE during the solvation process. Moreover, preliminary analyses indicate that the size-modified EASE functional surpasses its counterpart based on the classic PB theory across various analytic aspects. Our work is the first step toward calculating EASE through the utilization of diffuse-interface VISM.

This study presents a comprehensive mathematical framework that applies fluid dynamics to model the spatial spread of infectious diseases with low mortality rates. By treating susceptible, infected, and treated population densities as fluids governed by a system of partial differential equations, the study simulates the epidemic’s spatial dynamics. The Monotone Upwind Scheme for Conservation Laws is employed to enhance the accuracy of numerical solutions, providing a high-resolution approach for capturing disease transmission patterns. The model’s analogy between fluid flow and epidemic propagation reveals critical insights into how diseases disperse geographically, influenced by factors like human mobility and environmental conditions. Numerical simulations show that the model can predict the evolution of infection and treatment population densities over time, offering practical applications for public health strategies. Sensitivity analysis of the reproduction number highlights the influence of key epidemiological parameters, guiding the development of more efficient disease control measures. This work contributes a novel perspective to spatial epidemiology by integrating principles of fluid dynamics, aiding in the design of targeted interventions for controlling disease outbreaks.